Can a neuron have both a bias and a threshold?
I assume you're talking about a perceptron threshold function. One definition of it with an explicit threshold is
$$f(\textbf{x})= 
\begin{cases}
1&amp; \text{if } \textbf{w}\cdot\textbf{x} &gt; t\\
0&amp; \text{otherwise}
\end{cases}.$$

Another form with a bias is
$$f(\textbf{x})= 
\begin{cases}
1&amp; \text{if } \textbf{w}\cdot\textbf{x} + b &gt; 0\\
0&amp; \text{otherwise}
\end{cases}.$$

But these forms are of course equivalent if you set $b=-t$.

There's nothing stopping you from using a perceptron definition with both a bias and a threshold:
$$f(\textbf{x})= 
\begin{cases}
1&amp; \text{if } \textbf{w}\cdot\textbf{x} +b &gt; t\\
0&amp; \text{otherwise}
\end{cases}.$$

But this is also equivalent to the other two forms. We can rewrite this as
$$f(\textbf{x})= 
\begin{cases}
1&amp; \text{if } \textbf{w}\cdot\textbf{x} &gt; t'\\
0&amp; \text{otherwise}
\end{cases}$$
where $t'=t-b$ is the new threshold. Or, we could rewrite it as 
$$f(\textbf{x})= 
\begin{cases}
1&amp; \text{if } \textbf{w}\cdot\textbf{x} + a&gt; 0\\
0&amp; \text{otherwise}
\end{cases}$$
where $a=b-t$ is the new bias.

You never see definitions of this function with both a threshold and a bias because it has simpler forms.